Generalized Dehn Functions II

0.17 MB·

by  Chad Groft

GENERALIZED DEHN FUNCTIONS II
C.L.GROFT
9
0
0 Abstract. For G a group of type Fq, we establish the existence, finiteness,
2 and uniqueness up to scaling of various q-dimensional isoperimetric profiles.
n We also show that these profiles all coincide for q ≥ 4, and that significant
a overlap exists for q =3. When G has decidable wordproblem, this has mild
J consequences forthegrowthratesoftheseprofiles. Wealsoestablishametric
analogue forhighlyconnected Riemannianmanifolds.
5
1

]
R
Introduction
G
In part I (see [5]) we examined the isoperimetric profiles ΦX,M where M is
.
h an orientable manifold of dimension q ≥ 2 with nonempty boundary and X is
t
a either a CW complex or a local Lipschitz neighborhood retract (LLNR). Briefly,
m if f: ∂M → X is a map with volume v that extends to M, then f extends to
g: M →X withvolumeatmostΦX,M(v),andΦX,M(v)isthesmallestnonnegative
[
extended real number with this property. In order for a map g: M → X to have
1 a volume, it must be a map (M,∂M) → (X(q),X(q−1)) (which in [5] is called
v
quasi-cellular); there are several possible definitions in this case, as discussed in
7
1 [5]. When M = Dq, these profiles are identical to the higher-dimensional Dehn
3 functions defined in [1]. Similar profiles ΦX,q, for which we replace maps ∂M →X
2 andM →X with (q−1)-andq-dimensionalchains respectively,werealsodefined,
.
1 both in [5] and earlier in [3] and [6].
0 Unlike the more familiar Dehn function of a complex (think M = D2), the
9 e e
functions ΦX,M and ΦX,q (for X a finite complex) are not dependent on π (X)
0 1
alone; the homotopy groups π (X) through π (X) are also relevant. Moreover,
: 2 q−1
v e e
if the higher homotopy groups are nonzero, it is possible for ΦX,M and ΦX,q to
i
X be different functions, and ΦXe,M may depend nontrivially on M (this is not yet
r clear). The best theorem we have at present is the following: Given a continuous
a
function f: X →Y where f : π (X)→π (Y) is an isomorphismfor 1≤t<q, the
∗ t t
e e
functions ΦX,M and ΦY,M are quasi-equivalent; that is,
e e
ΦX,M(v)≤A·ΦY,M(Bv)+Cv+D
e e
for some constants A, B, C, D, and vice versa. The chain versions ΦX,q and ΦY,q
are also quasi-equivalent. Moreover, if X is a compact Riemannian manifold or
compact Lipschitz neighborhood retract (CLNR) with a triangulation, the profile
e
ΦX,M may be interpreted as applying to cellular maps or to Lipschitz maps; the
e
two interpretations are quasi-equivalent functions, and similarly for ΦX,q.
As in [1] (with M =Dq) and [2], given a finite CW complex or CLNR X where
X is (q − 1)-connected, one can define the isoperimetric profiles of G = π (X)
1
e
Date:January15,2009.
1
2 C.L.GROFT
to be those of X; that is, δM = ΦG,M = ΦXe,M and FVq = ΦG,q = ΦXe,q. The
G G
above results ensure that ΦG,M and ΦG,q are well-defined up to quasi-equivalence.
e
It remains to ask whether ΦG,M = ΦG,q, or whether ΦG,M = ΦG,N for N another
q-dimensional connected orientable manifold with nonempty boundary.
Our major result is that ΦG,M = ΦG,q almost everywhere for q ≥ 4 and M
a q-dimensional manifold, and that ΦG,M = ΦG,N for M and N manifolds of
dimensionq ≥3with∂M =∂N. Moregenerally,ΦX,M =ΦX,q almosteverywhere
and ΦX,M = ΦX,N under these conditions, provided X is (q −1)-connected. In
the cellular case, these equalities are exact; in the metric case, the second is exact,
whilethefirstmayfailatdiscontinuitiesofthefunctions(ofwhichthereareatmost
countablymany,sincebothfunctionsareincreasing). Thisgeneralizesatheoremin
[2], which states that ΦG,Mq ≤ ΦG,Dq provided q ≥ 4 and either ∂M is connected
or ΦG,Dq is superadditive, and which can be proved by itself by the methods of
lemma 1.
The proof uses, and is inspired by, a classic theorem of Brian White, proved in
[12]: Let X be a simply connected Riemannian manifold, let Mq be an orientable
connected manifold with boundary where q ≥ 3, and let f: M → X be Lipschitz.
Then the Plateau problem for Lipschitz maps g ≃ f rel ∂M is equivalent to the
Plateau problem for integral currents T ∼f ([M]).
♯
Since weare lookingatallfunctions whichfill agivenmapf: ∂M →X,andall
currents T where ∂T = f ([M]), this theorem is not quite sufficient. To finish the
♯
proof, we need for any integral current T a function f where f ([M]) is very close
♯
to T. We use the assumptionthatX is (q−1)-connected, the Hurewicz maps, and
the strong approximation theorem to prove that such a function must exist. As in
[12], we start with X a CW complex (a mild generalization, as [12] uses simplicial
complexes) and use this to prove the metric case.
Section 1 establishes the above argument in detail, allowing us in section 2 to
definetheisoperimetricprofilesforcertaingroupsGandestablishtheirequality. In
athirdsection,weshowthatthe profilesfora givengroupGarefinite everywhere,
and computably bounded provided G has solvable word problem.
Most of the concepts and conventions used herein were introduced in [5], and
the acknowledgements there apply here as well.
1. The map vs. current equivalence
Let X be a (q −1)-connected CW complex where q ≥ 3, and let M be a q-
dimensional compact orientable manifold with ∂M 6= ∅. We want to show that
ΦX,M ≤ ΦX,q, with equality in dimensions q ≥ 4. To do this, we must show that
cell ch
chains in dimensions q and q−1 can be represented by functions Mq−1 →X(q−1)
and Mq →Xq, where M is some compact manifold, possibly with boundary. The
easiest cases are where M = Dq or M = Sq−1; here we can interpret “volume” to
mean“wordlengthinπ (X(q),X(q−1))”orthe sameonedimensiondown. SinceX
q
is highly connected, the Hurewicz maps are length-preserving isomorphisms, and
thisissufficient. FormoregeneralM,wecantriangulateM andputallthevolume
on a single cell.
The metric case is similar, with one wrinkle. If T is a polygonal current in X
of dimension q, then T can be represented by a Lipschitz map f: M → X with
the same volume (by the above logic applied to some highly connected simplicial
complexwhichsupportsT). However,anintegralcurrentT withvolumev canonly
GENERALIZED DEHN FUNCTIONS II 3
beapproximatedbyapolygonalcurrent,withvolumeatmostv+ǫ. Thus,ifΦX,M
met
has a jump discontinuity at v, then ΦX,q(v) lies between ΦX,M(v) and the right
cur met
limit. As ΦX,M is an increasing function, this implies that ΦX,M = ΦX,q almost
met met cur
everywhere,and certainly the functions are quasi-equivalent.
To begin, consider the diagram
π (X(q),X(q−1)) ϕ // H (X(q),X(q−1))
q ∼ q
∂ ∂
(cid:15)(cid:15) (cid:15)(cid:15)
π (X(q−1)) ϕ // H (X(q−1))
q−1 (cid:127)_ ∼ q−1 (cid:127)_
j∗ j∗
(cid:15)(cid:15) (cid:15)(cid:15)
π (X(q−1),X(q−2)) ϕ //H (X(q−1),X(q−2))
q−1 q−1
where j: (X(q−1),∗) → (X(q−1),X(q−2)) is the inclusion, each ϕ is a Hurewicz
homomorphism, and each ∂ is the usual boundary map. By both parts of [10,
theorem7.4.3],thisdiagramcommutes. BytheHurewiczisomorphismtheorem,all
of the maps ϕ are isomorphisms if q ≥ 4, and the top and bottom maps preserve
word length. If q = 3, then the top two ϕ are still isomorphisms, the top ϕ still
preserves length, and the last ϕ is a length-reducing epimorphism. The j on the
∗
right is an injection, by the long exact sequence for (X(q−1),X(q−2)) (certainly
H (X(q−2)) = 0), so the j on the left is an injection as well for q ≥ 4. (The
q−1 ∗
left j is also aninjection if q =3 by the exactsequence ofhomotopy groups,since
∗
π (X(1))=0.) Note that the compositionj ◦∂ onthe rightis our previousnotion
2 ∗
of ∂ on chains; since j is 1-1, we might ignore this distinction.
∗
Lemma 1. Let q ≥ 3, let τ be a triangulation of M, and let X be a (q − 1)-
connected CW complex. Let f: ∂M → X be quasi-cellular, and let T be a q-chain
of X where ∂T = f ([∂M]). Then there exists a τ-cellular map g: M → X where
♯
∂g =f, g ([M])=T, and Vol g =kTk.
♯ τ
Proof. Recall that Vol g is the sum, over all q-cells ∆ of the triangulation τ, of
τ
the word length of [g ↾∆] in π (X(q),X(q−1)). In fact, we will construct g so that
q
everyq-cellexceptpossiblyoneissenttoX(q−1). LetG=(V,E)betheundirected
graph where V is the set of q-cells of τ, and where {v,w} ∈ E iff v and w share
a (q−1)-face. G is connected, so let R first be a spanning tree for G. We define
a procedure which will define g cell by cell on M, while removing cells from R.
At any given stage, g will be defined only on ∂M and on those cells which are no
longerin R. In particular,if {v,w}∈R at any stage,then g will not be defined on
the interior of v∩w, as this cell is neither a subset of ∂M nor a face of any q-cell
where g is already defined.
Proceed as follows: While R contains more than one vertex, let v be a leaf of R
and let w be the unique q-cell where {v,w} ∈ R. Let D = (∂v)\(v∩w)◦, which
is homeomorphic to Dq−1. There is some subcomplex of D on which g is already
defined (possibly empty). Since X(q−1) is (q−2)-connected, we may extend g in
some way to all of D. Finally, there is a retraction r: v → D; define g on v as
g◦r. (Note that g ([v])=g (r ([v]))=0.) Now that g is defined on v, remove the
♯ ♯ ♯
vertex v and the edge {v,w} from R. The remaining cells still form a connected,
etc. manifold, at least in the PL category, so we may repeat.
4 C.L.GROFT
When this procedure is finished, let ∆ be the unique q-cell remaining in R, so
that g is defined on M \∆◦. Since g ([v]) = 0 for every other q-cell v, we have
♯
g ([∂∆]) = ∂T. Choose h: (Dq,Sq−1) → (X(q),X(q−1)) where ϕ([h]) = T. Then
♯
there is a homotopy H: ∂h≃f. Attach H to h to obtain g ↾∆. (cid:3)
Theorem1. Letq ≥4and∂M 6=∅,andletX bea(q−1)-connectedCWcomplex.
Then ΦX,q =ΦX,M. If q =3, then ΦX,M ≤ΦX,q.
ch cell cell ch
Proof. Let q ≥ 3, let f: ∂M → X(q−1) be a quasi-cellular map, and suppose
Vol f ≤n. Thereisatriangulationτ ofM whereVol f =Vol f. BecauseX(q)
tr ∂τ tr
is (q−1)-connected, there is some extension of f to a τ-cellular map f′: M →X.
Thus the (q−1)-chain S = f ([∂M]) is the boundary of the q-chain f′([M]). Let
♯ ♯
T be a q-chain of least possible volume where ∂T = S. By lemma 1, there is a τ-
cellular map g: M →X where ∂g =f and Vol g = kTk≤ ΦX,q(n). Generalizing
τ ch
over all f, τ, and n, ΦX,M ≤ΦX,q.
cell ch
To see the reverse inequality for q ≥ 4, choose a triangulation τ on M and a
q-cell∆with atleastone face onthe boundary onM;callthis face δ. Lets be the
map on M which collapses all points of M, except those in ∆◦ or δ◦, to a single
point ∗; s is a map from M to Dq which is a diffeomorphism ∆◦ ∼= (Dq)◦ and
δ◦ ∼=(Sq−1\{∗}).
Given a (q−1)-boundary S =∂T where kSk≤n, let f′: Sq−1 →X(q−1) where
ϕ([f′]) = S and f′(∗) ∈ X(0) (this is where we use q ≥ 4). Let f = f′◦s: ∂M →
X(q−1), so that Vol f = kSk. f′ has a filling g′, by the Hurewicz isomorphism,
τ
so f has a filling g′◦s. Choose a filling g for f so that Vol g is minimum; then
tr
T =g ([M]) fills S and
♯
FV (S)≤kTk≤Vol g =FV f ≤ΦX,M(n).
ch tr cell cell
Generalizing over all S and n, ΦX,q ≤ΦX,M. (cid:3)
ch cell
Corollary 1. For q ≥4, X as above, and M, N manifolds with nonempty bound-
ary, ΦX,M =ΦX,N. This still holds true for q =3 provided ∂M ∼=∂N.
cell cell
Proof. The case q ≥ 4 follows directly from theorem 1; ΦX,M = ΦX,q = ΦX,N.
cell ch cell
For q =3, refer to the first paragraph of the above proof; by this reasoning, every
quasi-cellular function f: (∂M = ∂N) → X(q−1) has a filling on both M and N,
and moreover FVM f =FV f ([∂M])=FVN f. Thus ΦX,M =ΦX,N. (cid:3)
cell ch ♯ cell cell cell
Now we address the metric case. Let X be an LLNR; that is, let X ⊆U ⊆RN
where U is open in RN, and let r: U →X be a locally Lipschitz retraction.
Theorem2. Letq ≥3,∂M 6=∅,X a(q−1)-connectedLLNR,P afinitesimplicial
complex of dimension q, and ψ: P → X a 1-1 Lipschitz map. Let T = ψ ([P]).
♯
Then there exists f: M → X where f ([M]) = T and Vol f = M(T). If q ≥ 4,
♯ met
one may choose f so that Vol ∂f =M(∂T).
met
Proof. Construct Q ⊃ P so that Q is q-dimensional and (q−1)-connected. (For
example, find a high-dimensional simplex which contains P as a subcomplex and
take its q-skeleton.) Extend ψ to a continuous map ψ′: Q → X; this is possible
because X is (q−1)-connected. Mollify ψ′ outside P to obtain a Lipschitz map
ψ′′: Q → U, and let ψ′′′ = r ◦ψ′′. Then ψ′′′ is a Lipschitz map Q → X which
extends ψ. For convenience, refer to ψ′′′ as ψ.
GENERALIZED DEHN FUNCTIONS II 5
Asintheproofoftheorem1,chooseatriangulationτ ofM,acell∆intersecting
∂M,andacollapsingmaps: (M,∂M)→(Dq,Sq−1)whichis adiffeomorphismon
∆◦ and(∆∩∂M)◦. Chooseg: (Dq,Sq−1)→(Q(q),Q(q−1))whereϕ([g])=[P]. We
may assume that g covers each point in the interior of a q-cell of P exactly once,
thatitdoessosmoothly,andthatg[Dq]doesnotintersectthe interiorofanyother
q-cell of Q; and similarly for ∂g provided q ≥4. Let f =ψ◦g◦s. Then
f ([M])=ψ (g ([Dq]))=ψ ([P])=T.
♯ ♯ ♯ ♯
Over every cell of τ except for ∆, the volume form of f∗(ds2) disappears, and we
calculate
Vol f =Vol (ψ◦g)= ψ∗(ds2)=M(T),
met met Z
P
the last because ψ is 1-1. A similar calculation shows that Vol ∂f =M(∂T) for
met
q ≥4. (cid:3)
A similar result holds for q ≥ 3 and ∂M = ∅, provided ∂T = 0. (The only way
this can happen is if ∂P = ∅ as well, which by exactness implies P = j (P′) for
∗
some P′ ∈Hq−1(Q(q−1))∼=πq−1(Q(q−1)).)
Theorem 3. Let q ≥4 and let M and X be as in theorem 2. Then
ΦX,M ≤ΦX,q ≤ΦX,M.
met cur met
In particular, ΦX,M =ΦX,q almost everywhere. If q =3, then ΦX,M ≤ΦX,q.
met cur met cur
Recall that f is the upper envelope of f, where f is any map from a topological
space to R.
Proof. [12, theorem 3] tells us that, given a function f: M →X,
inf{Vol g :g ≃f rel ∂M}=inf{M(T):T −f ([M])∈B (X)}.
met ♯ q
Letq ≥3,r≥0,andf ∈C0,1(∂M,X),Vol f ≤r. SinceX is(q−1)-connected,
met
f is the boundary of some continuous map h: M → X, and h can be mollified
and retracted to a Lipschitz map M →X. Thus S =f ([∂M]) is the boundary of
♯
h ([M]).
♯
Let R∈I (X) where M(R)<FV S+ǫ. R−h ([M]) is a q-cycle, which is a
q cur ♯
q-boundaryup to an elementof Hq(X)(singular homology). But Hq(X)∼=πq(X),
so by modifying h in a disk, we may assume R−h ([M]) is a boundary. Thus
♯
FV f ≤inf{Vol g :g ≃h rel ∂M}=inf{M(T):T −h ([M])∈B (X)}
met met ♯ q
≤M(R)<ΦX,q(r)+ǫ.
cur
Taking ǫ→0 and generalizing over all f and r, we have ΦX,M ≤ΦX,q.
met cur
Conversely, for q ≥ 4 let S ∈ I (X) where ∂S = 0 and M(S) ≤ r. By the
q−1
strong approximation theorem [4, lemma 4.2.19], S is homologous by a current R
where M(R) < ǫ to a polyhedral current S′ = ψ ([P]), M(S′) < M(S)+ǫ. Note
♯
that ∂S′ = 0 as well. By theorem 2, choose f: ∂M → X where f (∂M]) = S′
♯
and Vol f = M(S′). f extends to some g: M → X; choose g where Vol g <
met met
FV (f)+ǫ. Then
met
FV S <M(g ([M]))+ǫ<FV (f)+2ǫ≤ΦX,M(r+ǫ)+2ǫ;
cur ♯ met met
taking ǫ → 0, FV S ≤ ΦX,M(r) (since ΦX,M is increasing). Generalizing over S
cur met met
and r, we have ΦX,q ≤ΦX,M.
cur met
6 C.L.GROFT
A function can differ from its upper envelope only at points of discontinuity;
since ΦX,M is increasing, there are at most countably many of these (see [9]), so
met
ΦX,M =ΦX,q almost everywhere. (cid:3)
met cur
Corollary 2. For q ≥ 4, X as in theorem 2, and M and N manifolds with
nonempty boundary, ΦX,M = ΦX,N almost everywhere. Provided ∂M ∼= ∂N, the
met met
conclusion holds for q ≥3 and with exact equality.
Proof. Thefirstfollowsfromtheorem3. Forthesecond,followthereasoninginthe
first paragraph of the preceding proof. Every f ∈ C0,1(∂M,X) has a filling, and
FVM f =FV f ([∂M])=FVN f. Therefore ΦX,M =ΦX,N. (cid:3)
met cur ♯ met met met
The case q = 3 deserves some attention. For simplicity, assume that we only
consider M where ∂M is connected; thus ∂M = Σ is the surface of genus g for
g
some g ≥ 0. By Corollary 2, only ∂M is relevant, so assume M = Γ is the
g
solid torus of genus g. If g ≤ h, then every map Γ → X can be composed with
g
a collapsing map Γ → Γ to produce a new map with equal volume and filling
h g
volume. Thus ΦX,Γg ≤ ΦX,Γh and ΦX,Γg ≤ ΦX,Γh everywhere. As every 2-chain
cell cell met cell
or polygonal 2-current T can be represented by a map f: Σ → X for some g, we
g
haveΦX,3 =lim ΦX,Γg everywhereand ΦX,3 =lim ΦX,Γg almosteverywhere.
ch g↑0 cell cur g↑0 met
As we will note later, there are spaces X for which ΦX,D3 6= ΦX,Γ1. Such a
cell cell
separationbetween ΦX,Γg and ΦX,Γh for 1≤g <h, or betweenΦX,Γg and ΦX,3, is
cell cell cell ch
not yet known.
2. Applications to geometric group theory
We say that a group G is of type F if there is a CW complex X = K(G,1)
q
where X(q) is finite. Equivalently, G is of type F iff there is a finite CW-complex
q
Y whereY is (q−1)-connectedandπ (Y)=G. Y may be takenasq-dimensional;
1
alternatively, Y may be a compact manifold.
e
For example, every group is of type F . A group is of type F iff it is finitely
0 1
generated, and of type F iff it is finitely presented. Every type F is a strict
2 q
subtype of F ; this follows in the case q =3 by the results in [11].
q−1
Inthiscase,whereX =K(G,1)hasfinite q-skeleton,wemaytakethefunctions
ΦXe,q and ΦXe,M (for M a q-manifold) as invariants of G. For example, ΦXe,D2 is
ch cell cell
the classical Dehn function, while ΦXe,Dq is the higher-order Dehn function δ
cell q−1
studied in [1]. As in these cases (and for similar reason), if we change X to a
different K(G,1), we obtain a quasi-equivalent function. By the results of the last
section, many of these functions coincide.
Lemma 2. Let q ≥ 2, and let G be a group of type F . Let X and Y be CW
q
complexes which are K(G,1)’s and where X(q) and Y(q) are both finite. Then
e e e e
ΦX,q ≈ΦY,q. Also ΦX,M ≈ΦY,M for M q-dimensional and ∂M 6=∅.
ch ch cell cell
Proof. Letϕ: π (X,∗)→π (Y,∗)beanisomorphism. By[7,Theorem1B.9],there
1 1
is a continuous function f: X →Y where f =ϕ. The conclusions follow from [5,
∗
theorem 2]. (cid:3)
Definition 1. Let q ≥ 2. Let G be a group of type F , and let X be a K(G,1)
q
where X(q) is finite. The chain isoperimetric profile of G in dimension q is that of
GENERALIZED DEHN FUNCTIONS II 7
X;
e
e ΦGch,q :=ΦXch,q.
Likewise, for M a q-manifold with ∂M 6= ∅, the cellular isoperimetric profile of G
for M is that of X;
e
e ΦGce,lMl :=ΦXce,llM.
For example, if G is finitely presented, the function ΦG,D2 is defined, and is
cell
in fact the usual Dehn function of G. As one would expect, this definition is an
abuse of language, since ΦG,q and ΦG,M are defined only up to quasi-equivalence.
ch cell
It still makes sense to ask whether these functions are linear, or polynomial, or
exponential, or computably bounded, or everywhere finite.
Theorem 4. Let G is a group of type F , where q ≥ 4. Then for any q-manifold
q
M with ∂M 6=∅, ΦG,M ≈ΦG,q. If q =3, then ΦG,M 4ΦG,q; and if M and N are
cell ch cell ch
two 3-manifolds with ∂M =∂N 6=∅, then ΦG,M ≈ΦG,N.
cell cell
Proof. Apply theorem 1 and corollary 2 to any K(G,1) with finite q-skeleton. (cid:3)
Theorem 5. Let X be a closed Riemannian manifold where π (X)=G and X is
1
(q−1)-connected. Then ΦXceu,rq ≈ΦGch,q, and ΦXmee,Mt ≈ΦGce,lMl for any Mq. e
Proof. Choosea triangulationτ onX. There is a K(G,1)whose q-skeletonis τ(q);
one adds (q+1)-cells to kill π , etc. Thus
q
ΦXe,q ≈Φτe,q ≈ΦG,q and ΦXe,M ≈Φτe,M ≈ΦG,M. (cid:3)
cur ch ch met cell cell
3. Finiteness and computability
OneoftheclassicresultsonDehnfunctionsoffinitelypresentedgroupsGisthat
δ is recursive, or even subrecursive, iff the word problem on G is solvable. The
G
prooffromrighttoleftisfairlysimple: FixX =K(G,1)withfinite 2-skeleton. Up
totranslationbyG,thereareonlyfinitelymanyloopsinX whoselengthisatmost
n, for any fixed n. Ofthese, use the wordproblemsolutionto determine whichare
e
contractible, then search all the disk maps into X (up to translation) until one is
found for each contractible loop.
e
The situation for generalized Dehn functions is not so clear-cut. For example,
a result by Papasoglu (see [8]) states that the “second Dehn function” ΦG,D3 is
cell
subrecursiveforall groupsGoftypeF ,whileanunpublishedresultbyYoung(see
3
[13]) shows that there is a group G, which is of type F for all q, where ΦG2,Γ1 is
q cell
notsubrecursive,noris ΦGq−1,q for any q ≥4. In fact, the secondDehn function is
ch
effectivelytheonlygeneralizedDehnfunctionwhichissubrecursiveforallqualifying
G.
Nevertheless, results in one direction is possible.
Theorem 6. Let q ≥2 and let G be a group of type F . Then ΦG,q(n) is finite for
q ch
all n. If G has solvable word problem, then ΦG,q is recursive.
ch
TheproofisfairlysimilartothatfortheclassicalDehnfunction,withthewrinkle
that the class of chains with volume at most n is not generally finite, even up to
translation by G. We observe, however, that each chain can be decomposed into
connected components, and the class of small connected chains can be exhausted.
8 C.L.GROFT
TheequivalentofΦG,q whereonlyconnectedboundariesareconsideredistherefore
ch
computable. If the components of a cycle are far enough apart, then the most
efficient way to fill the cycle is to fill each component separately. This allows us to
compute ΦG,q in terms of the specialized version.
ch
Theorem 7. Let q ≥ 2, and let G be a group of type F , and let M be a q-
q
dimensional manifold with nonempty boundary. Then ΦG,M(n) is finite for all n.
cell
If G has solvable word problem, then ΦG,M is subrecursive.
cell
For q ≥3,the [recursive]bound onΦG,M is precisely ΦG,q; for q =2,the bound
cell ch
may be computed in terms of the classical Dehn function.
We need some preliminaries. Let X be a CW-complex. For any t and for any
t-chains A, B, we say B is a subchain of A if kAk=kBk+kA−Bk. Equivalently,
for ever t-cell σ of X, let n and n be the coefficients of σ in the expansions
A,σ B,σ
of A and B respectively. Then B is a subchain of A iff 0 ≤ n ≤ n or
B,σ A,σ
n ≤n ≤0 for all σ. A given chain A has finitely many subchains.
A,σ B,σ
We say B is a component of A if B is a subchain of A and ∂B is a subchain
of ∂A, and that a t-chain A is connected if its only components are 0 and A. By
inductiononkAk,everychainAcanbeexpressedasasumofconnectedcomponents
A = B +···+B . Given such an expansion, we have kAk = kB k+···+kB k
1 n 1 n
and k∂Ak=k∂B k+···+k∂B k. Thus, if ∂A=0, then ∂B =0 for all i.
1 n i
Proof of theorem 6. Let X be a K(G,1) with finite q-skeleton, and let X be its
universalcover. Assume G is finite, so that X(q) is finite. For 0≤t≤q and v ≥0,
e
we can determine the set of all elements T ∈ C (X) where kTk ≤ v. Also we can
e t
determine whether a given T ∈C (X) is a cycle, hence a boundary. Thus we have
t e
the following algorithm for ΦG,q(n): determine all the (q−1)-chains S which are
ch e
boundaries; for each N starting with 0 and every q-chain T where kTk = N, flag
∂T; stop when every S is flagged and output N.
Now assume G is infinite. Each cell of X is coveredby a collection of cells of X
in 1-1correspondence with the elements of G, andthe collectionis invariantunder
e
the natural G-action. For each t ≤ q, let Σ be a set containing exactly one t-cell
t
of X which covers any given t-cell of X; thus Σ is finite.
t
Up to G-action, there are finitely many connected t-chains of any fixed volume
e
n. For n = 0, this is obvious. For n ≥ 1, suppose A is a connected t-chain and
kAk = n. We generate chains B , B , ..., B = A, each a subchain of the next,
1 2 n
wherekB k=k,asfollows: AftertranslationbysomeelementofG,Amustcontain
k
±σ where σ ∈ Σ . Let B = ±σ. Suppose B has been chosen for some k < n.
t 1 k
Since A is connected, B is not a component of A, so there is some subchain C of
k
A−B where kCk = 1 and k∂(B +C)k < k∂B +∂Ck. Let B = B +C.
k k k k+1 k
There are only finitely many possibilities for B ; and for any chain B, there are
1
onlyfinitelymanycellswhichshareboundarywithB,soforanychoiceofB there
k
are finitely many choices for B . Thus there are finitely many possible B =A,
k+1 n
which is what we wanted.
Let Ψ(n)=maxFV (A), where the maximum is taken over connected (q−1)-
ch
cycles A with kAk≤n. Let
Φ(n)= max Ψ(k),
partitionsP ofnX
k∈P
GENERALIZED DEHN FUNCTIONS II 9
where P is interpreted as a multiset. Ψ(n) and Φ(n) are finite for all n.
e e
We claim that ΦX,q(n) = Φ(n) for all n. To see ΦX,q(n) ≤ Φ(n), let A be
ch ch
a (q −1)-cycle with kAk ≤ n. Let A = B +···+B be a sum of connected
1 m
components. Each B is a cycle; let C ∈ C (X) with minimum volume where
i i q
∂C =B . The multiset {kB k,...,kB k} is a partition of kAk, so
i i 1 m e
m m m
FV (A)≤ kC k= FV (B )≤ Ψ(kB k)≤Φ(kAk)≤Φ(n).
ch i ch i i
X X X
i=1 i=1 i=1
e e
Take the supremum over all A to see ΦX,q(n) ≤ Φ(n). In particular, ΦX,q(n) is
ch ch
finite for all n.
e
To see Φ(n) ≤ ΦX,q(n), let B , ..., B be a finite list of connected (q −1)-
ch 1 m
cycles where kB k≤n. Choose translates B′ =g B as follows: Let g =e and
i i i i i 1
B′ = B . FoPr i = j +1, note that there are finitely many connected q-chains C
1 1
e
where∂C sharesa cellwith any ofB′, ..., B′ and where kCk≤ΦX,q(n). Since G
1 j ch
is infinite, there must be some h ∈G where h = B does not share a cell with ∂C
i
for any such C. Take g to be some such h. Note that kB′k=kB k, ∂B′ =0, and
i i i i
FV (B′)=FV (B ) for all i.
ch i ch i
Let A=B′ +···+B′ , so that ∂A=0 and kAk≤n. Let C be a q-chainwhere
1 m
∂C = A and kCk = FV (A). If C′ 6= 0 is a connected component of C, then ∂C′
ch
e
shares cells with B′ for a unique i, since kC′k ≤kCk ≤ ΦX,q(n) and we chose the
i ch
B′ so that no such connected q-chain exists with boundary cells from two distinct
i
B′. Conversely, any boundary cells of ∂C′ must come from ∂C = A = B′,
i i i
and if ∂C′ = 0, then C −C′ is a filling cycle for A of smaller volume, wPhich is
impossible. Thus, given a sum of connected components C =C +···+C , each
1 N
C is associatedwitha unique B′,andC is thereforeasumoverioffillings for B′.
j i i
Hence
m
e
FV (B′)=FV (A)≤ΦX,q(n).
ch i ch ch
X
i=1
e
Taking the supremum over all finite lists B , ..., B , we have Φ(n)≤ΦX,q(n), as
1 m ch
desired.
Finally, suppose G is infinite with solvable word problem (let Y be a finite set
of generators). We first show that Ψ is computable. If w ∈ F(Y) is a word and
σ ∈ Σ for t ≤ q, then (w,σ) represents the t-cell wσ. Every t-cell is represented
t
by some pair, and (w,σ) and (w′,σ′) represent the same cell iff σ =σ′ and w−1w′
representsthe trivialelement ofG. Chains A∈C (X)canbe representedaslinear
t
combinationsofpairs. FromarepresentationforA,onecancalculatekAk;also,one
e
cancalculatea representationfor ∂A. Also, one candetermine allthe subchains of
A, and one can decide whether a chain A is connected: for each subchain B where
B 6= 0 and B 6= A, determine whether ∂B is a subchain of ∂A. A is connected iff
this is never the case.
The proof that Ψ is computable is similar to the case with G finite. Generate
representations of all of the (q −1)-chains A with kAk ≤ n, up to G-action, by
starting with cells in Σ and adding cells which share boundary components;
q−1
there are finitely many such chains. Of these chains, take the subset consisting of
connected cycles. For each connected cycle A, generate the q-chains B where ∂B
and A share at least one cell, in order of increasing volume, stopping when a chain
10 C.L.GROFT
B with ∂B =A is found; remember the volume of B. The maximum such volume,
across all A, is Ψ(n).
e
Given an algorithm for computing Ψ, computing Φ=ΦX,q is easy. (cid:3)
Proof of theorem 7. For q ≥ 3 the theorem is an easy consequence of theorems 1
and 6. Now consider q = 2. As noted previously, the classical Dehn function of
G, here ΦG,D2, is finite for all inputs, and that it is computably bounded (even
cell
computable) iff G has solvable word problem. For arbitrary M2, we have ∂M a
disjoint union of k copies of S1, which implies
k
ΦG,M(n)≤ max ΦG,D2(n ).
cell n1+···+nk=nX cell i
i=1
Similarly, if A is a 1-cycle,it can be represented as A +···+A where each A is
1 k i
the image of a copy of S1; hence
ΦG,2(n)≤ max ΦG,D2(k). (cid:3)
ch cell
P apartitionofnX
k∈P
4. Further questions
The relationship between ΦG,2 and the various ΦG,M for dimM = 2 is not yet
ch cell
clear. Also we do not yet know whether ΦG,M is always recursive for dimM ≤ 3
and G decidable, rather than subrecursive. Finally, there is no obviousconverse to
the results of section 3; that is, if G does not have decidable word problem, it is
unclear what restrictions this places on ΦG,q or ΦG,M.
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